Effect of small impurities on the superconducting transition temperature

Volume 2, number 5

THE

PHYSICS

EFFECT OF SUPERCONDUCTING

LETTERS

1 October 1962

SMALL IMPURITIES ON TRANSITION TEMPERATURE

D. S. K O T H A R I , V. S. M A T H U R and N. P A N C H A P A K E S A N Physics Department, University of Delhi, India Received 11 September 1962

R is now well recognised 1-3) that the presence of small impurities in superconductors results in a lowering of the transition temperature T c. For superconductors with small impurity content, i.e., when the electronic m e a n free path I is greater than the coherence distance ~o, the change in T c is proportional to 1//. Furthermore, this change is almost the s a m e for all superconductors 2), and is also independent of the nature of the impurities. In the case of very dirty superconductor s, however, i.e., when l < ~o, the change in T c becomes c o m plicated and is no longer independent of the nature of impurities. A theory of dirty superconductors has been given by Anderson 4), according to which if the impurities are numerous, i.e., if I < ~o, then the Cooper pair should not be constructed with the usual Bloch eigenstates of opposite momenta, but by pairing each electron state obtained in the presence of scatterers, with its time reverse. If, however, the impurities are small in n u m b e r so that I > ~o, it is a s s u m e d that the impurities scatter the Cooper pair itself. In the presence of impurities, since translational invariance does not hold, a pair need not conserve m o m e n t u m * on scattering. Between two scatterings, however, the m o m e n t u m K of the pair is preserved. W h e n averaged over a large time or when averaged over a large n u m b e r of pairs at a given time, the m e a n value of Kvanishes but the average magnitude IKI is ~ 1/I, as required by the uncertainty principle. For calculating the correlation energy, it is reasonable to a s s u m e that for l > ~o a pair does not have zero m o m e n t u m but a m o m e n t u m IKI ~ 1/I. The purpose of this note is to investigate how far this idea of Cooper pairs with non-zero m o m e n t u m can provide s o m e understanding to the problem of superconductors with dilute impurities. For a pair of finite m o m e n t u m IKI, the available phase space for scattering is smaller than when IKI = 0. If N(IKI) be the phase space per unit

* See L. N. Cooper 5) who has used this concept in the problem of the Knight shift.

v o l u m e p e r u n i t e n e r g y a t t h e F e r m i s u r f a c e , then f o r s m a l l IKI

N(IKI)=N(O)

(1

1

IK1

2km_ko- ] ,

(1)

w h e r e k m - k o (~ 10 6 c m "1) i s t h e r e g i o n in m o mentum space around the Fermi momentum kF, r e s p o n s i b l e f o r s u p e r c o n d u c t i v i t y . If w e a s s u m e t h a t t h e e l e c t r o n - p h o n o n i n t e r a c t i o n V(0) i s not s e n s i b l y c h a n g e d t in t h e p r e s e n c e of s m a l l i m p u r i t y c o n t e n t , t h e e n e r g y gap ~( IKI ) i s g i v e n by 1

e ( I K I ) = 2(~¢0)av e N ( ] K ] ) V ( O )

(2)

The c h a n g e in t h e t r a n s i t i o n t e m p e r a t u r e T c i s given b y ATc Tc

_ A{

c

~

Z~J(IKI)

(3)

N2(0) Y(0)

An e s t i m a t e of t h i s c h a n g e can be m a d e b y p u t t i n g z1, so t h a t f r o m e q s . (1) and (3), w e g e t

NV~

ATc 2 1 T c -km - ko l "

(4)

The c h a n g e in t r a n s i t i o n t e m p e r a t u r e A T c d e p e n d s o n l y on t h e m e a n f r e e p a t h , and i s i n d e p e n d e n t of t h e n a t u r e of i m p u r i t i e s d e t e r m i n i n g t h e f r e e p a t h . F u r t h e r m o r e , it f o l l o w s f r o m t h e i s o t o p e e f f e c t t h a t T c / ( k m - ko) p r o p o r t i o n a l to T c / 0 D i s a c o n s t a n t , so t h a t A T c i s a l s o i n d e p e n d e n t of t h e n a t u r e of t h e s u p e r c o n d u c t o r . Chanin et a l . 2) h a v e e x p e r i m e n t a l l y d e t e r m i n e d t h e s l o p e of t h e A T c v e r s u s 1// c u r v e , and o u r r e s u l t f r o m eq. (4) i s in r o u g h agreement with theirs. A precise numerical comp a r i s o n w i l l not be v e r y m e a n i n g f u l , in v i e w of the s o m e w h a t q u a l i t a t i v e n a t u r e of o u r a r g u m e n t s . S i n c e t h e l o w e r i n g of T c i s i n d e p e n d e n t of t h e w a y in w h i c h t h e m e a n f r e e p a t h i s d e t e r m i n e d , one would e x p e c t eq. (4) to hold even when t h e m e a n f r e e p a t h i s i n t r o d u c e d b e c a u s e of s c a t t e r i n g f r o m t h e b o u n d a r i e s , a s in t h e c a s e of thin s u p e r c o n d u c t i n g f i l m s . The m e a n f r e e p a t h l in eq. (4) t See, however, ref. 6). See also refs. 7, 8). 235

Volume 2, number 5

PHYSICS

w i l l n o t b e e x a c t l y e q u a l to t h e t h i c k n e s s of t h e f i l m , b u t w i l l b e s o m e w h a t l a r g e r a s shown b y D i n g l e 10). R e c e n t l y , Lynton a n d M c L a c h l a n 11) h a v e i n d e e d o b s e r v e d s u c h an e f f e c t , and t h e y f i n d t h a t t h e d e c r e a s e in T c i s in e x c e l l e n t a g r e e m e n t w i t h t h e e x p e r i m e n t s of C h a n i n e t a l . 2) on i m p u r e s a m p l e s . T h i s i s a l s o in a g r e e m e n t w i t h eq. (4). On t h e b a s i s of t h e s e c o n s i d e r a t i o n s , one would e x p e c t v e r y thin f i l m s ( t h i c k n e s s l e s s t h a n ~o) to c o r r e s p o n d to " d i r t y 't s u p e r c o n d u c t o r s , a n d , t h e r e f o r e , e x p e c t a s a t u r a t i o n of t h e e f f e c t on T c. A n o t h e r e f f e c t t h a t m i g h t b e of i m p o r t a n c e in t h e c a s e of v e r y thin s u p e r c o n d u c t o r s i s the e f f e c t of s u r f a c e t e r m s on t h e e n u m e r a t i o n of t h e w a v e f u n c t i o n s f o r a c o n t i n u u m 12). A s i m p l e c a l c u l a t i o n s h o w s t h a t t h i s w i l l n o t b e of i m p o r t a n c e u n l e s s the f i l m s a r e v e r y thin * (~ 10 A). One of u s (V.S.M.) i s g r a t e f u l to t h e C o u n c i l of • It is interesting to note that this effect accounts well for the change in transition temperature of thin liquid He4 films (see ref. 13)).

LETTERS

1 October 1962

S c i e n t i f i c and I n d u s t r i a l R e s e a r c h f o r f i n a n c i a l help.

References 1) E.A. Lynton, B.Serin and M. Zucker, J. Phys. Chem. Solids 3 (1957) 165. 2) G. Chanin, E . A . Lynton and B. Serin, Phys. Rev. 114 (1959) 719. 3) R.I. Gayley J r . , E.A.Lynton and B.Serin, Phys. Rev. 126 (1962) 43. 4) P.W. Anderson, J. Phys. Chem. Solids 11 (1959) 26. 5) L.N. Cooper, Phys. Rev. Letters 8 (1962) 367. 6) A. B. Pippared, J. Phys. Chem. Solids 3 (1957) 175. 7) D.J.Kenworthy and D . t e r Haar, Phys. Rev. 123 (1961) 1181. 8) D. J. Kenworthy, M.J. Zuckermann, D.M. Brink and D. ter Haar, Physics Letters 1 (1962) 35. 9) J. Bardeen, L.N. Cooper and J. R. Schrieffer, Phys. Rev. 108 (1957) 1175. 10) R.B. Dingle, Proc. Roy. Soc. (London) A 201 (1950) 545. 11) E.A. Lynton and D.McLachlan, Phys. Rev. 126 (1962) 40. 12) D. S. Kothari and F.C.Auluck, Nature 159 (1947) 204. 13) A.D.Singh and R.K. Pathria, Nature 183 (1959) 668.

* * * * *

THERMAL

EXPANSION

OF

PYROLYTIC

GRAPHITE

Miss F. ENTWISLE United Kingdom Atomic Energy Authority, Reactor Materials Laboratory, Culcheth, Warrington, Lancs. Received 8 September 1962

Thermal expansion coefficients over three temp e r a t u r e r a n g e s ( - 1 9 6 o c to 20oc, 20oc to 120oc, 100°C to 700°C) h a v e b e e n m e a s u r e d on s p e c i m e n s of p y r o l y t i c g r a p h i t e . T h i s m a t e r i a l i s v e r y h i g h l y o r i e n t a t e d a n d s p e c i m e n s w e r e cut in o r d e r to g i v e an i n d i c a t i o n of t h e t h e r m a l e x p a n s i o n of t h e s i n g l e c r y s t a l in t h e c a n d a d i r e c t i o n s . The pyrolytic graphite was obtained from High temperature materials , Inc., U.S.A., where it was p r o d u c e d b y d e p o s i t i o n of g r a p h i t e f r o m t h e g a e o u s p h a s e on to a c a r b o n s u b s t r a t e a t 2200oc. The g r a p h i t e w a s h e a t t r e a t e d in a f u r n a c e in an a r g o n a t m o s p h e r e a t 2800°C f o r 8 h o u r s b e f o r e t h e r m a l e x p a n s i o n m e a s u r e m e n t s w e r e m a d e in a c o n v e n t i o n a l s i l i c a d i l a t o m e t e r t u b e 1). T h e 0.5 c m d i a m e t e r s p e c i m e n s w e r e 2.6 c m and 0.5 c m l o n g in t h e a a n d c d i r e c t i o n s , r e s p e c t i v e l y . F o r t h e two l o w e r t e m p e r a t u r e r a n g e s , c h a n g e s in l e n g t h of t h e g r a p h i t e s p e c i m e n s w e r e m e a s u r e d b e t w e e n two f i x e d temperatures using a Mercer precimeter graduated in d i v i s i o n s of 0.000 02 in. F o r t h e r a n g e up to

236

7 0 0 o c t h e d i l a t o m e t e r w a s h e a t e d a t a r a t e of 1 . 3 o c p e r m i n u t e in a f u r n a c e in an a r g o n a t m o s p h e r e . T e m p e r a t u r e , m e a s u r e d b y a t h e r m o c o u p l e on t h e s p e c i m e n , and c h a n g e in l e n g t h , m e a s u r e d b y a 10 c m d i a m e t e r M e r c e r d i a l g a u g e g r a d u a t e d in d i v i s i o n s of 0.0002 c m , w e r e o b s e r v e d a t 30°C i n t e r v a l s . A s d i f f i c u l t y h a s b e e n e x p e r i e n c e d in m e a s u r i n g a d e c r e a s e in l e n g t h of a s p e c i m e n in a dilatometer tube over the highest temperature range, the a direction specimen was stacked with a g r a p h i t e s p e c i m e n of known c o e f f i c i e n t of t h e r m a l e x p a n s i o n so a s to g i v e a r e s u l t a n t i n c r e a s e in l e n g t h with t e m p e r a t u r e . In t h e c d i r e c t i o n two p y r o l y t i c s p e c i m e n s w e r e s t a c k e d t o g e t h e r to p r o duce a higher total expansion over all three temp e r a t u r e r a n g e s . M e a s u r e m e n t s w e r e r e p e a t e d and t h e r e s u l t s w e r e a l l w i t h i n + 5% of t h e v a l u e s given l a t e r in f i g s . 1 - 4. A c o r r e c t i o n w a s a p p l i e d f o r t h e e x p a n s i o n of t h e s i l i c a d i l a t o m e t e r t u b e ; t h e c o e f f i c i e n t of t h e r m a l e x p a n s i o n of s i l i c a w a s e s t i m a t e d f r o m I n t e r n a t i o n a l C r i t i c a l T a b l e s 2) to b e